Functional Calculus Theory for Incompressible Fluid Turbulence

A functional integral representation for the space-time Hopf characteristic functional is derived from the probability theory for a statistical ensemble of velocity fields that satisfy the Navier-Stokes equation for boundary-free incompressible fluid flow. The functional integral representation involves a pair of real vector field integration variables denoted by u and v, and the evaluation of the integral is performed in two steps. First, the integration over the field variable u is effected exactly in the general case by applying methods of explicit functional integration. Second, the resulting functional integral over the field variable v is reduced to a form amenable to specialized analysis by applying a suitable transformation of the integration field variable v → z. Specializing to mathematically defined ``C-dominant turbulence,'' the final functional integration over the field variable z is effected exactly and yields a characteristic functional of Gaussian form. The two-point velocity correlation tensor for C-dominant turbulence is then obtained from the characteristic functional.